Properties

Label 2-280-280.237-c1-0-5
Degree $2$
Conductor $280$
Sign $-0.703 + 0.710i$
Analytic cond. $2.23581$
Root an. cond. $1.49526$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.00402 + 1.41i)2-s + (−1.09 + 1.09i)3-s + (−1.99 + 0.0113i)4-s + (0.721 + 2.11i)5-s + (−1.55 − 1.54i)6-s + (−2.63 − 0.264i)7-s + (−0.0241 − 2.82i)8-s + 0.595i·9-s + (−2.99 + 1.02i)10-s − 2.35i·11-s + (2.18 − 2.20i)12-s + (−1.27 + 1.27i)13-s + (0.363 − 3.72i)14-s + (−3.11 − 1.53i)15-s + (3.99 − 0.0455i)16-s + (−1.60 + 1.60i)17-s + ⋯
L(s)  = 1  + (0.00284 + 0.999i)2-s + (−0.633 + 0.633i)3-s + (−0.999 + 0.00569i)4-s + (0.322 + 0.946i)5-s + (−0.634 − 0.631i)6-s + (−0.994 − 0.100i)7-s + (−0.00853 − 0.999i)8-s + 0.198i·9-s + (−0.945 + 0.325i)10-s − 0.711i·11-s + (0.629 − 0.636i)12-s + (−0.354 + 0.354i)13-s + (0.0972 − 0.995i)14-s + (−0.803 − 0.395i)15-s + (0.999 − 0.0113i)16-s + (−0.389 + 0.389i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.703 + 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.703 + 0.710i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.703 + 0.710i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.208364 - 0.499557i\)
\(L(\frac12)\) \(\approx\) \(0.208364 - 0.499557i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.00402 - 1.41i)T \)
5 \( 1 + (-0.721 - 2.11i)T \)
7 \( 1 + (2.63 + 0.264i)T \)
good3 \( 1 + (1.09 - 1.09i)T - 3iT^{2} \)
11 \( 1 + 2.35iT - 11T^{2} \)
13 \( 1 + (1.27 - 1.27i)T - 13iT^{2} \)
17 \( 1 + (1.60 - 1.60i)T - 17iT^{2} \)
19 \( 1 + 2.48iT - 19T^{2} \)
23 \( 1 + (-0.615 + 0.615i)T - 23iT^{2} \)
29 \( 1 - 0.914T + 29T^{2} \)
31 \( 1 - 5.53iT - 31T^{2} \)
37 \( 1 + (7.02 - 7.02i)T - 37iT^{2} \)
41 \( 1 + 5.48iT - 41T^{2} \)
43 \( 1 + (-7.03 - 7.03i)T + 43iT^{2} \)
47 \( 1 + (8.82 - 8.82i)T - 47iT^{2} \)
53 \( 1 + (-4.92 - 4.92i)T + 53iT^{2} \)
59 \( 1 - 5.85iT - 59T^{2} \)
61 \( 1 - 1.78T + 61T^{2} \)
67 \( 1 + (-5.48 + 5.48i)T - 67iT^{2} \)
71 \( 1 + 10.6T + 71T^{2} \)
73 \( 1 + (-2.44 - 2.44i)T + 73iT^{2} \)
79 \( 1 - 8.51iT - 79T^{2} \)
83 \( 1 + (6.43 - 6.43i)T - 83iT^{2} \)
89 \( 1 + 8.47T + 89T^{2} \)
97 \( 1 + (-13.8 + 13.8i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.62009724389054899323262666632, −11.24183931239000677538520076204, −10.39306561637895660280825028885, −9.705487576317150077309319567455, −8.653076637147217797924802520082, −7.26223195573428194927845721507, −6.45461667627746173878093832491, −5.67408325498249460911143763479, −4.46276204212709611909727683571, −3.13467626416293569004473283445, 0.43433089791539338018898612534, 2.06008735719514852250263742968, 3.74466204766978232512923171023, 5.10107234797730427022548028586, 6.03002008106654843623737858120, 7.32730328408895046251362289123, 8.733064742599799737515228849818, 9.565260773281333090269029177168, 10.23716962511974300754634979143, 11.62913620338064462156779826471

Graph of the $Z$-function along the critical line